Crossed products by semigroups of endomorphisms and the Toeplitz algebras of ordered groups

Adji, Sriwulan; Laca, Marcelo; Nilsen, May; Raeburn, Iain

doi:10.1090/S0002-9939-1994-1215024-1

Crossed products by semigroups of endomorphisms and the Toeplitz algebras of ordered groups
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by Sriwulan Adji, Marcelo Laca, May Nilsen and Iain Raeburn

Proc. Amer. Math. Soc. 122 (1994), 1133-1141

DOI: https://doi.org/10.1090/S0002-9939-1994-1215024-1

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Abstract:

Let ${\Gamma ^ + }$ be the positive cone in a totally ordered abelian group $\Gamma$. We construct crossed products by actions of ${\Gamma ^ + }$ as endomorphisms of ${C^ \ast }$-algebras, and give criteria which ensure a given representation of the crossed product is faithful. We use this to prove that the ${C^\ast }$-algebras generated by two semigroups V, $W:{\Gamma ^ + } \to B(H)$ of nonunitary isometries are canonically isomorphic, thus giving a new, self-contained proof of a theorem of Murphy, which includes earlier results of Coburn and Douglas.

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